Permutation and Combination Calculator

What this tool does

Permutation and Combination Calculator computes nPr (permutations, order matters, no repeats), nCr (combinations, order doesn’t matter, no repeats), and the two repetition variants: n^r for permutations where repeats are allowed, and (n+r−1)Cr for combinations where repeats are allowed. Every result shows the factorial formula behind it.

Permutations and combinations answer the same underlying question, how many ways can you select or arrange r items from a set of n, but the exact count depends on two separate choices: does order matter, and can an item be chosen more than once. Mixing up which of the four resulting formulas applies is the single most common combinatorics mistake.

This calculator covers all four cases directly: nPr for arrangements where order matters and nothing repeats, nCr for selections where order doesn’t matter and nothing repeats, n^r for arrangements where repeats are allowed, and the combinations-with-repetition formula (n+r−1)Cr for selections where repeats are allowed. Every result shows the exact factorial formula used, computed with exact integer (BigInt) arithmetic so even large n and r produce an exact, not approximated, answer.

How to use Permutation and Combination Calculator

1. Choose a Calculation: permutations, combinations, permutations with repetition, or combinations with repetition.
2. Enter n, the total number of items available.
3. Enter r, the number of items being chosen or arranged.
4. Read the resulting count and the factorial formula used to compute it.
5. Switch modes to compare how the count changes depending on whether order matters and whether repeats are allowed.

Tips for better results

• If arranging contestants on a podium (order matters, no repeats), use permutations; if simply picking a committee from a group (order doesn’t matter, no repeats), use combinations.
• Permutations with repetition is the right formula for passwords or PINs where the same character can appear in multiple positions, such as a 4-digit PIN with n = 10 possible digits.
• nCr always equals nPr divided by r!, since choosing r items and then separately arranging them in every possible order recovers every permutation exactly once.
• For large n and r, the counts grow extremely fast; this calculator uses exact BigInt arithmetic rather than floating-point numbers so results stay exact rather than rounding at very large values.

Common use cases

• Working out how many ways a set of runners can finish in the top three positions (a permutation problem).
• Calculating how many different 5-card hands can be dealt from a 52-card deck (a combination problem).
• Counting how many possible 4-digit PINs exist, where digits can repeat (permutations with repetition).
• Teaching the four combinatorics cases side by side with the same n and r to see how dramatically the count changes.

Common mistakes to avoid

• Using nPr when the problem actually doesn’t care about order, which overcounts every valid selection by r! because it also counts every reordering of the same group.
• Forgetting that nPr and nCr require r ≤ n, since you cannot choose or arrange more items than exist without allowing repetition.
• Confusing the combinations-with-repetition formula with simple combinations; they produce very different counts once repeats are genuinely allowed.